Difference between revisions of "Tyler Houlihan - Difference equations - a few examples with Partial Fraction Expansion explanation ECE301Fall2008mboutin" - Rhea

Revision as of 10:28, 24 October 2008

Definition

A system characterized by a difference equation in DT is given as:

$ \, \sum_{k=0}^N a_k\,y[n-k]=\sum_{k=0}^N b_k\,x[n-k] $

We will likely be asked to solve for the frequency response $ \,H(e^{j\omega}) $, the unit impulse response $ \,h[n] $, or the system's response to an input $ \,x[n] $.

Example 1

Find $ \,H(e^{j\omega}) $, and $ \,h[n] $ for the following system in DT domain:

$ \, \frac{2}{5}y[n-1]-\frac{3}{5}y[n-2]+6y[n]=4x[n] $

Solution

First find $ \,H(e^{j\omega}) $:

1) Take the fourier transform of every term:

$ \, \frac{2}{5}\mathcal{Y}(\omega)e^{-j\omega}-\frac{3}{5}\mathcal{Y}e^{-2j\omega}+6\mathcal{Y}(\omega)=4\mathcal{X}(\omega) $

2) Factor out the y terms:

$ \, \mathcal{Y}(\omega) \left (\frac{2}{5}e^{-j\omega}-\frac{3}{5}e^{-2j\omega}+6 \right )=4\mathcal{X}(\omega) $

3) Now isolate $ \,H(\omega) $

$ \, H(\omega)=H(e^{j\omega})=\frac{\mathcal{Y}(\omega)}{\mathcal{X}(\omega)}=\frac{4}{\frac{2}{5}e^{-j\omega}-\frac{3}{5}e^{-2j\omega}+6} $

2nd, find $ \,h[n] $

$ h[n]=\mathcal{F}^{-1}(H(e^{j\omega}) $

This is the rough part, as partial fraction expansions must be used :P

for simplification purposes, let $ \,x=e^{-j\omega} $ , so the fraction becomes :

$ \, \frac{4}{\frac{2}{5}x-\frac{3}{5}x^2+6} $